There are triangle-free $d$-degenerate graphs with chromatic number $d+1$; see http://dx.doi.org/10.1006/jctb.1999.1910 or 
http://dx.doi.org/10.1017/S0963548399004022 or
https://arxiv.org/pdf/1310.2972.pdf. So I suspect that the desired characterisation is impossible.